Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential

Abstract

We consider the Schrödinger operator Lα on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner-von Neumann potential c sin(2ωx+δ)x γ , where γ ∈ ( 1 2 , 1). The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point νcr is an eigenvalue of the operator Lα for some value of the boundary parameter α = αcr, specific to that particular point. We prove that for α = αcr the spectral density of the operator Lα has a zero of the exponential type at νcr.

Titchmarsh-Weyl formula for the spectral density of a class of Jacobi matrices in the critical case

“…Several papers were devoted to establishing such formulas in both discrete and continuous cases, [16,19,26]. That analysis has been used to study the behavior of the spectral density of discrete [27] and differential [22,28] Schrödinger operators with the Wigner-von Neumann potential near the critical points which appear due to that form of the potential. These formulas were derived for special classes of operators, and, moreover, for the non-critical case.…”

Embedded eigenvalues and Neumann–Wigner potentials for relativistic Schrödinger operators

“…Bound states corresponding to positive eigenvalues have been realized also experimentally [6]. Since this initial example, Neumann-Wigner type potentials have attracted much attention [35,38,14,27,5,9,16,4,3,25,26,37]. In particular, the set of embedded eigenvalues is not necessarily a small set, Simon has shown that examples can be constructed for which there is a dense set of positive eigenvalues [36], see also [29,31,33].…”

Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique

“…Although it is not proved here, we expect that the spectral density (i.e. the derivative of the spectral function) vanishes with power-like decay at points where we have subordinate solutions, giving pseudogaps in the spectrum (see [26] and [31] for the continuous case, and [30] for the discrete Schrödinger operator case). The potentials involved here to obtain these subordinate solutions have a Wigner-von Neumann structure (see, for example, [16, 20-22, 26, 28, 29]) which in its original conception for Schrödinger equations had the form c sin(2ωx + ϕ)…”